Very many mathematical notions (groups, rings, algebras, etc) form categories.

Similarly, a number of mathematical notions (spaces, spectra, derived schemes, etc.) form $(\infty,1)$-categories. Their study is what modern algebraic topology is all about.

To summarize:
Modern algebraic topology is the study of everything that forms an $(\infty,1)$-category.

Very many mathematical notions (groups, rings, algebras, etc) form categories.

Similarly, a number of mathematical notions (spaces, spectra, derived schemes, etc.) form $(\infty,1)$-categories. Their study is what modern algebraic topology is about.

To summarize:
Modern algebraic topology is the study of everything that forms an $(\infty,1)$-category.